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ISEE Upper Level Math : Numbers and Operations

Study concepts, example questions & explanations for ISEE Upper Level Math

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Multiply Exponents

Simplify:

$(6a^3)(\frac{1}{3}a^2)$

Possible Answers:

2a^6

2a^5

a^6

a^5

2a^4

Correct answer:

2a^5

Explanation:

Based on the product rule for exponents in order to multiply two exponential terms with the same base, add their exponents:

$(x)^a(x)^b=x^{a+b}$

So we can write:

$(6a^3)\left(\frac{1}{3}a^2\right)=\left(\frac{6}{3}\right)\times a^{3+2}=2a^5$

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Example Question #4 : How To Multiply Exponents

Evaluate:

$\frac{4x^2(x^3)^{3}}{8x^7}$

Possible Answers:

$\frac{1}{2}x^5$

$\frac{1}{2}x^3$

x^4

$\frac{1}{2}x^4$

x^3

Correct answer:

$\frac{1}{2}x^4$

Explanation:

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. In addition, based on the product rule for exponents, in order to multiply two exponential terms with the same base we need to add their exponents:

$(x)^a(x)^b=x^{a+b}$

So we can write:

$\frac{4x^2(x^3)^{3}}{8x^7}=\frac{4x^2(x^{3\times 3})}{8x^7}=\frac{4x^2\times x^9}{8x^7}=\left(\frac{4}{8}\right)\frac{x^{2+9}}{x^7}=\frac{1}{2}\cdot \frac{x^{11}}{x^7}$

in order to divide two exponents with the same base, we can keep the base and subtract the powers. So we get:

$\frac{4x^2(x^3)^{3}}{8x^7}=\frac{1}{2}\cdot \frac{x^{11}}{x^7}=\frac{1}{2}x^{11-7}=\frac{1}{2}x^4$

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Example Question #4 : How To Multiply Exponents

Evaluate:

$-2x^2(x^{-3})^{3}$

Possible Answers:

$-2x^{7}$

$-2x^{-7}$

$2x^{7}$

$-x^{-7}$

$2x^{-7}$

Correct answer:

$-2x^{-7}$

Explanation:

Based on the negative exponent rule we have:

$a^{-n}=\frac{1}{a^n}$

which says negative exponents in the numerator get moved to the denominator and become positive exponents. And negative exponents in the denominator get moved to the numerator and become positive exponents. So we can write:

$-2x^2(x^{-3})^{3}=-2x^2\left(\frac{1}{x^3}\right)^{3}$

In addition, based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. We also know that when a fraction is raised to a power, the numerator and the denominator are both raised to that power. So we can write:

$-2x^2(x^{-3})^{3}=-2x^2\left(\frac{1}{x^3}\right)^{3}=-2x^2\times \frac{1^3}{x^{3\times 3}}$

$=-2x^2\times \frac{1}{x^9}$

$=\frac{-2x^2}{x^9}$

$=-2x^{2-9}$

$=-2x^{-7}$

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Example Question #21 : Exponents

If x=2 , find the value of:

$\frac{3x^{\sqrt{2}}(x^{\sqrt{3}})^\sqrt{3}}{x^{\sqrt{2}}}$

Possible Answers:

Correct answer:

Explanation:

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. So we can write:

$\frac{3x^{\sqrt{2}}(x^{\sqrt{3}})^\sqrt{3}}{x^{\sqrt{2}}}=3\times \left(\frac{x^{\sqrt{2}}}{x^{\sqrt{2}}}\right) \times (x^{\sqrt{3}})^{\sqrt{3}}=3x^{\sqrt{3}\times \sqrt{3}}=3x^3$

Substitute x=2 and we get:

$3x^3=3\times 2^3=3\times 8=24$

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Example Question #4 : How To Multiply Exponents

Simplify:

$(x^{2\sqrt{2}})^{3\sqrt{2}}$

Possible Answers:

$x^{6\sqrt{}2}$

$x^{6}$

$x^{8}$

$x^{12}$

$x^{2}$

Correct answer:

$x^{12}$

Explanation:

Based on the power rule, we know that in order to raise a power to a power we need to multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ .

$(x^{2\sqrt{2}})^{3\sqrt{2}}=x^{2\sqrt{2}\times 3\sqrt{2}}=x^{2\times 3\times 2}=x^{12}$

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Example Question #9 : How To Multiply Exponents

Simplify:

$(\frac{12x^{3}}{3x^{-1}})^{4}$

Possible Answers:

$16x^{16}$

$256x^{16}$

$256x^{8}$

$256x^{12}$

$16x^{8}$

Correct answer:

$256x^{16}$

Explanation:

The Negative Exponent Rule says $a^{-n}=\frac{1}{a^n}$ .

$(\frac{12x^{3}}{3x^{-1}})^{4}=(\frac{12}{3}\times x^{3+1})^{4}=(4x^4)^{4}$

The power rule says that, in order to raise a power to a power, we need to multiply the exponents, i.e. $(x^{m})^{n}=x^{m\times n}$ .

$(\frac{12x^{3}}{3x^{-1}})^{4}=(4x^4)^{4}=4^4\times x^{4\times 4}=256x^{16}$

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Example Question #71 : Numbers And Operations

What is the value of this equation?

$(x^{2})^{3}-(2^{3})^{2}$

Possible Answers:

$x^{6}-32$

$x^{5}-64$

$x^{6}+64$

$x^{6}-64$

Correct answer:

$x^{6}-64$

Explanation:

When an exponent is raised to another exponent, the exponents should be multiplied toghether. This will result in:

$(x^{2})^{3}-(2^{3})^{2}$

$x^{6}-2^{6}$

$x^{6}-64$

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Example Question #72 : Numbers And Operations

Which expression is equal to $(x^{2})^{3}$ ?

Possible Answers:

$x^{\frac{3}{2}}$

$x^{5}$

$x^{\frac{2}{3}}$

$x^{6}$

Correct answer:

$x^{6}$

Explanation:

When exponent of a value is raised to another exponent, the values of the exponents are multiplied by each other.

$(x^{2})^{3}$

2 is multiplied by 3, and so the exponent of x is 6.

$x^{6}$

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Example Question #73 : Numbers And Operations

What is the value of $(2^{6})^{\frac{1}{2}}$ ?

Possible Answers:

Correct answer:

Explanation:

When one exponent is raised to another exponent, the values of the exponents should be multiplied together. Thus,

$(2^{6})^{\frac{1}{2}}$ can be simplified to $2^{3}$ , given that $6\cdot \frac{1}{2}=3$ .

$2^{3}=2\cdot2\cdot2=8$

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Example Question #74 : Numbers And Operations

What is the expression below equal to?

$3x^{4}*5x^{9}$

Possible Answers:

$8x^{36}$

$15x^{36}$

$15x^{13}$

$8x^{9}$

Correct answer:

$15x^{13}$

Explanation:

When exponents are multiplied by each other, the powers should be added together. Meanwhile, numbers not raised to an exponent are simply multiplied by each other.

Therefore, the answer is $15x^{13}$ , because 3*5=15 , and 9+4=13 .

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Numbers and Operations

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Multiply Exponents

Example Question #4 : How To Multiply Exponents

Example Question #4 : How To Multiply Exponents

Example Question #21 : Exponents

Example Question #4 : How To Multiply Exponents

Example Question #9 : How To Multiply Exponents

Example Question #71 : Numbers And Operations

Example Question #72 : Numbers And Operations

Example Question #73 : Numbers And Operations

Example Question #74 : Numbers And Operations

All ISEE Upper Level Math Resources

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